Revisiting the Quantum Geometry of Torus-fibered Calabi-Yau Threefolds

TL;DR

This paper demonstrates that the modularity properties of topological string amplitudes on torus-fibered Calabi-Yau threefolds are equivalent to a wave-function property under conifold monodromy, connecting Gromov-Witten invariants with Donaldson-Thomas indices.

Contribution

It establishes the equivalence between modularity of topological string amplitudes and a wave-function property under conifold monodromy, introducing a holomorphic, modular covariant variant of the partition function.

Findings

Modularity properties follow from wave-function behavior under conifold monodromy.

The generating series of GV invariants are quasimodular, matching mock-modular behavior of D4-D2-D0 indices.

Analyzed and tabulated CY threefolds with N-sections up to N=5, including new examples.

Abstract

About ten years ago, Katz, Klemm and Huang conjectured that topological string amplitudes on compact, elliptically fibered Calabi-Yau threefolds at fixed base degree could be expressed in terms of meromorphic Jacobi forms for $SL(2,\mathbb{Z})$, giving access to Gromov-Witten invariants at arbitrary genus. This was later generalized to torus-fibered CY threefolds with $N$-sections, where topological string amplitudes are conjecturally governed by meromorphic Jacobi forms under the congruence subgroup $\Gamma_1(N)$. In this work, we show that these modularity properties follow from (and are equivalent to) the wave-function property of the topological string partition function $Z_{\rm top}$ under a relative conifold monodromy, implementing a particular Fourier-Mukai transformation on the derived category of coherent sheaves. In particular, we introduce a variant of $Z_{\rm top}$ which is both holomorphic and modular covariant. Under the same relative conifold monodromy, the generating series of genus 0 Gopakumar-Vafa invariants at fixed base degree is mapped to the generating series of rank 0 Donaldson-Thomas indices counting D4-D2-D0-brane bound states wrapped on the torus fiber. We show that the quasimodularity of the generating series of GV invariants matches the expected mock-modular behavior of the generating series of D4-D2-D0 indices, despite having different multi-cover contributions. We analyze and tabulate a large number of CY threefolds fibered over del Pezzo surfaces, with an $N$-section for $N\leq 5$, including several new examples beyond the realm of toric geometry.